Quantum Sensing With GKP States: How Researchers Are Engineering Noise-Resistant Measurements

Now imagine that the signal is so small that ordinary measurement strategies struggle to distinguish it from noise.

This is one of the fundamental challenges in precision measurement.

In quantum physics, the problem becomes particularly interesting because uncertainty is not simply the result of imperfect equipment. Quantum systems have intrinsic statistical properties, and measurement strategies must work within the limits imposed by quantum mechanics.

In a paper published on April 1, 2026, in PRX Quantum, researchers Lautaro Labarca, Sara Turcotte, Alexandre Blais, and Baptiste Royer investigated a different approach to this problem. Their study, "Quantum Sensing of Displacements with Stabilized Gottesman-Kitaev-Preskill States," describes a proposed sensor that uses quantum states known as Gottesman-Kitaev-Preskill states, or GKP states.

The researchers developed a protocol for estimating displacement along two independent directions in quantum phase space. Their design uses continuous stabilization to protect the sensor state while extracting information about the unknown signal.

The study reports two particularly important results:

• Theoretical analysis shows that the proposed sensing protocol can approach the multivariate quantum Cramér–Rao bound, a fundamental benchmark for estimation precision.
• Numerical simulations indicate that the protocol can surpass the Gaussian limit for displacement sensing with prior information, even when realistic noise is included in the model.

The proposed sensor can also operate continuously without a full reset between measurements, making it potentially useful for tracking signals that change over time.

There is an important scientific distinction: this paper presents a theoretical protocol and numerical simulations. It does not report a new laboratory demonstration of this complete sensor.

Nevertheless, the work offers a compelling example of how ideas from quantum error correction can be repurposed to improve measurement.

A displacement is a shift in the state of a system.

In classical mechanics, it might mean moving a mechanical component a tiny distance. In quantum physics, displacement can also describe a change in the state of a harmonic oscillator.

A harmonic oscillator is a system that behaves mathematically like a mass attached to a spring. Its quantum description includes two complementary quadratures, commonly labeled q and p. These are analogous to position-like and momentum-like coordinates, although their precise interpretation depends on the physical system and normalization.

Together, the quadratures describe a point in the oscillator's phase space.

A displacement changes the oscillator's phase-space coordinates. If a force acts on a trapped ion, for example, it may alter the ion's motional state. If an electromagnetic signal interacts with a resonator, it may similarly shift the oscillator's quantum state.

The sensing task is to estimate the unknown displacement from measurements of the resulting system.

Mathematically, an oscillator displacement can be represented by the displacement operator:

where ↠and â are the oscillator's creation and annihilation operators, and α is a complex parameter describing the displacement.

The real and imaginary components of α correspond to the two phase-space directions, subject to the chosen convention.

The challenge is that the unknown displacement may be very small. A sensor must distinguish it from uncertainty in the initial state, disturbances during the measurement, and errors in the final readout.

That is where specially engineered quantum states become useful.

Quantum squeezing is one established way to improve certain measurements.

A squeezed state redistributes uncertainty between two complementary quadratures. The uncertainty in one quadrature is reduced while the uncertainty in the other increases, consistent with the quantum uncertainty relation.

If the sensor needs to estimate one known quadrature, squeezing the relevant direction can improve precision.

But the 2026 study addresses a more demanding problem: estimating both quadratures of an unknown displacement.

The two-dimensional problem cannot simply be treated as two unrelated single-quadrature measurements without considering their joint measurement constraints.

There is also a practical issue. Many conventional strategies use Gaussian quantum states, such as coherent or squeezed states. Gaussian states have phase-space distributions with Gaussian mathematical forms and are comparatively well understood and accessible in many experimental platforms.

However, the best possible strategy is not necessarily Gaussian.

The researchers investigate a non-Gaussian alternative based on GKP states. These states have a grid-like structure in phase space and connect quantum sensing with the techniques used in bosonic quantum error correction.

The goal is not to eliminate quantum uncertainty. It is to engineer the state and measurement process so that the unknown displacement can be estimated with greater precision, even when noise is present.

Gottesman-Kitaev-Preskill states are commonly called GKP states, or grid states.

The concept was introduced by Daniel Gottesman, Alexei Kitaev, and John Preskill in a 2001 paper on encoding quantum information in an oscillator.

Unlike a qubit encoded in a pair of discrete energy levels, a bosonic system can encode information in the states of a harmonic oscillator, such as the motion of an ion or the electromagnetic field in a cavity.

The challenge is that an oscillator has many possible states and can be affected by small displacements. Those displacements can corrupt encoded information.

GKP encoding addresses this problem by arranging the quantum state into a periodic grid structure in phase space. Small errors can then be identified and corrected using the grid's structure.

An ideal GKP state involves infinitely sharp features and unbounded energy, so it cannot be prepared perfectly in a real laboratory. Practical implementations use finite-energy approximations.

This distinction matters for sensing. The theoretical grid is an idealized reference; an actual sensor must work with an imperfect state that has finite energy and is affected by decoherence.

The special state used in this research

The researchers focus on a particular grid state called the qunaught state.

The qunaught is not a conventional encoded logical qubit. Instead, it is a grid state used to stabilize the oscillator and support the sensing protocol.

This is important because the research is not simply about using an existing GKP-encoded qubit to perform a computation. It uses the stabilizing properties of a bosonic grid state to extract information about an unknown displacement.

The proposed method is designed to use the same process that stabilizes the state to gather information about the signal.

That connection between stabilization and measurement is one of the paper's central ideas.

The researchers build on a stabilization technique known as the small-big-small, or sBs, protocol.

The basic concept is to use controlled interactions between an auxiliary qubit and a bosonic mode. The bosonic mode stores the oscillator state, while the auxiliary qubit helps control and measure it.

A practical platform could be a superconducting circuit in which a qubit interacts with a microwave cavity, or a trapped-ion system in which an ion's internal spin couples to its motion.

The proposed protocol has several stages.

Stage one: Prepare the grid state

The oscillator begins in an initial state, which is driven toward a GKP-like grid state through repeated controlled operations.

The goal is to create a phase-space structure that can help the sensor distinguish a small displacement from ordinary fluctuations.

Stage two: Stabilize the oscillator

The protocol uses controlled displacements, qubit measurements or resets, and feedback operations to steer the oscillator toward the desired state.

This resembles a feedback-controlled system, but the relevant operations act on quantum states.

The stabilization process suppresses deviations from the desired grid structure. It is inspired by reservoir engineering, a technique in which a system's interaction with a carefully designed environment drives it toward a chosen state.

Stage three: Allow the unknown displacement to be encoded

The unknown displacement changes the oscillator's state.

Rather than treating this as a one-time disturbance followed by an entirely separate measurement, the researchers incorporate information extraction into the stabilization dynamics.

The auxiliary qubit produces measurement outcomes that depend on the state of the oscillator and therefore carry information about the displacement.

Stage four: Continue monitoring and updating

The protocol uses repeated measurements and feedback to continue stabilizing the state while accumulating information about the signal.

This allows the proposed sensor to operate continuously, without requiring a complete reset after each measurement cycle.

The researchers' method is therefore more than a proposal to prepare a sensitive state. It is a measurement strategy that attempts to maintain a useful state while extracting information from it.

In quantum mechanics, measurement backaction refers to the way a measurement affects the system being measured.

Backaction is not always harmful. Measurement can be used deliberately to prepare a state or drive a system toward a desired condition.

However, in precision sensing, the measurement process can also interfere with the signal or prevent continuous tracking of a changing quantity.

The authors describe their proposed device as a backaction-evading sensor.

In this context, the term means that the protocol is designed so that the measurement process does not simply destroy the sensor state and require a full reset before the next measurement. Instead, stabilization helps return the system toward its useful operating state.

This property is particularly relevant for signals that arrive continuously or vary over time.

Imagine monitoring a weak, changing force. A sensor that must repeatedly be taken offline, reset, and prepared again may lose information during those intervals. A sensor that can continuously estimate the signal while preserving its operating state could offer a practical advantage.

That does not mean the protocol eliminates every form of disturbance. Its performance still depends on the system's finite-energy state, the noise model, control accuracy, and the quality of the measurement process.

The paper's result is more specific: it presents a way to combine continuous GKP-state stabilization with displacement estimation in a protocol designed to avoid the usual need for a complete reset.

How do researchers determine whether a quantum sensor is close to optimal?

One important tool is the quantum Cramér–Rao bound.

In statistical estimation, the Cramér–Rao bound places a lower limit on the uncertainty of an estimated parameter under specified conditions. Quantum estimation theory extends this idea to measurements of quantum states.

For a single unknown parameter, the quantum Fisher information helps quantify how much information a quantum state can carry about that parameter.

For two-quadrature displacement sensing, the problem becomes more complex because researchers want to estimate two parameters together. The precision of joint estimation must account for the relationship between the parameters and the compatibility of the measurements used to estimate them.

The multivariate quantum Cramér–Rao bound provides a benchmark for this problem.

The researchers show that their protocol approaches this bound for two-quadrature displacement sensing.

This is a significant theoretical result: it means that the proposed strategy can come close to a fundamental precision benchmark within the framework analyzed in the paper.

But it does not mean that the sensor has unlimited precision, nor that a real device automatically achieves the bound. The result must be understood alongside the assumptions about the state, the measurement process, and the resources available.

Finite-energy grid states are imperfect. Noise can disturb the oscillator, qubit control can be imperfect, and measurements can introduce errors. The study therefore also examines performance beyond the idealized, noise-free situation.

The paper's second major result comes from numerical simulations.

The researchers simulate the sensing protocol under realistic noise assumptions and report that it can surpass the Gaussian limit for displacement sensing with prior information.

To understand this, it is important to distinguish three different benchmarks.

• Standard quantum limit: A commonly used precision benchmark whose scaling depends on how independent probes or resources are counted.
• Gaussian limit: A benchmark for a specified class of Gaussian-state sensing strategies. In this paper, the comparison includes prior information about the displacement.
• Multivariate quantum Cramér–Rao bound: A theoretical benchmark for the precision of joint estimation of multiple parameters.

These are not interchangeable terms. A protocol can beat one benchmark without proving that it beats every possible strategy under every possible resource constraint.

The researchers' numerical analysis indicates that their GKP-based approach can outperform the stated Gaussian benchmark even when realistic noise is included in the model.

That matters because non-Gaussian quantum states can be difficult to create and maintain. A theoretical advantage is less useful if it disappears as soon as noise is introduced.

The numerical result suggests that stabilization can help preserve an advantage under the modeled conditions.

However, numerical simulations are not experimental measurements. They evaluate the behavior of a mathematical model, using specified assumptions and approximations. A laboratory implementation may encounter additional sources of error or different operating constraints.

The next step is to implement the protocol physically, characterize the noise, and compare measured sensitivity against both the theoretical prediction and relevant Gaussian baselines.

Noise is one of the most important issues in practical quantum sensing.

A state that performs extremely well in a perfect mathematical model may become much less useful when exposed to real experimental conditions.

In a bosonic sensor, possible problems include:

• Motional heating or oscillator noise: The oscillator acquires unwanted energy or changes state.
• Decoherence: The quantum state loses useful phase relationships.
• Imperfect control: The displacement operations, rotations, or feedback do not exactly match their intended values.
• Finite-energy effects: The prepared GKP state is only an approximation to the ideal grid state.
• Readout errors: The auxiliary qubit's measurement result may be incorrect.
• Finite measurement time: A sensor may need to accumulate information over a limited interval.

The paper's simulations examine the resilience of the proposed protocol to noise. They do not establish that every listed error source is fully eliminated or that the sensor will perform identically in all experimental platforms.

The central lesson is that quantum error correction and stabilization may be useful not only for protecting stored information, but also for preserving the sensitivity of a measurement.

That is an important connection between two fields that are often discussed separately.

Displacement sensing is a foundational task in several areas of physics.

A force can cause a physical system to move or change its state. By measuring the displacement, researchers may be able to infer the force that produced it.

The relationship depends on the system's physical dynamics, so a more sensitive displacement measurement does not automatically translate into the same improvement in force sensitivity. The sensor must be calibrated, and its response to the force must be understood.

Nevertheless, the proposed GKP protocol could be relevant to future force-sensing instruments.

A second possible application is waveform estimation.

A waveform is a signal that changes over time. Examples include a time-varying force, an electric field, or a signal that moves through a detector.

Continuous sensing is valuable when a signal's time dependence carries important information. The ability to stabilize a sensor while monitoring the signal could help reduce the need for repeated preparation cycles.

The authors also identify quantum-channel learning as a potential application. This field investigates how to characterize unknown quantum processes, including how they transform quantum states.

These are proposed research directions, not applications already demonstrated by the paper.

A rigorous laboratory report should separate its conclusions according to the evidence behind them.

What the researchers established theoretically

The paper develops a protocol for two-quadrature displacement estimation using stabilized GKP states.

Its theoretical analysis shows that the proposed protocol can approach the multivariate quantum Cramér–Rao bound and explains how stabilization can be integrated with measurement.

What the researchers demonstrated numerically

The authors provide numerical simulations showing that the protocol can surpass the Gaussian limit for displacement sensing with prior information, including under the noise assumptions they model.

What remains experimentally unverified by this paper

The paper does not report a new laboratory demonstration of the complete proposed sensor.

It does not demonstrate a new gravitational-wave detector, a commercial precision instrument, or a dark-matter discovery.

It also does not show that every quantum sensor can overcome noise or that quantum error correction automatically improves every measurement.

Those conclusions would require additional evidence.

The value of the research is that it provides a specific protocol and a quantitative theoretical and numerical basis for testing it experimentally.

The next stage is to determine whether the proposed performance survives the constraints of real hardware.

A convincing experimental program would include several steps.

1. Prepare and stabilize a finite-energy GKP state. Researchers would need to establish that the chosen platform can create and maintain the required state with sufficient fidelity.

2. Apply known displacements. The team could introduce calibrated displacements in both phase-space quadratures and test whether the protocol accurately estimates them.

3. Measure performance over time. Because continuous operation is a major feature of the protocol, the sensor should be evaluated while monitoring a time-dependent signal rather than only through isolated measurements.

4. Quantify noise and control overhead. A meaningful performance comparison should include preparation time, stabilization cycles, feedback operations, readout errors, and the effects of finite energy.

5. Compare against appropriate baselines. The experimental data should be compared with Gaussian sensing strategies under equivalent assumptions about prior information, noise, measurement time, and available resources.

6. Test a real application. Only after characterizing the sensor should researchers determine whether it improves a specific force-measurement, waveform-estimation, or quantum-channel-learning task.

This final stage is especially important. A theoretical improvement in a carefully defined sensing problem does not automatically guarantee an advantage in every application.

The 2026 PRX Quantum study by Labarca and colleagues presents a compelling approach to quantum displacement sensing.

By combining GKP grid states, continuous stabilization, and auxiliary-qubit measurements, the researchers develop a protocol designed to estimate two quadratures of an unknown displacement while maintaining a useful quantum state.

Their theoretical analysis shows sensitivity approaching the multivariate quantum Cramér–Rao bound. Numerical simulations further indicate that the protocol can surpass the Gaussian limit for displacement sensing with prior information, even under modeled realistic noise.

The study's most important contribution is the connection between quantum error correction and quantum metrology. Stabilization is not used solely to preserve a quantum state; it becomes part of the measurement strategy itself.

That approach could prove useful for force sensing, continuous waveform monitoring, and learning unknown quantum processes.

The results remain a theoretical and numerical advance rather than a completed experimental sensor. The next milestone is to test the protocol in real hardware and determine whether its predicted advantages survive finite energy, imperfect control, and environmental noise.

The larger lesson is that progress in quantum technology does not always come from building a larger quantum computer. Sometimes it comes from learning how to engineer a fragile quantum system so that it can measure the physical world more effectively.

Read more in my related articles, Quantum Entanglement Explained, Quantum Superposition Explained, Long-Lived Quantum States in Ytterbium Ions, and The Optical Magnus Effect.